Write each of the following expressions as a single logarithmic term:
\log_{10} 5 + \log_{10} 4
\log_{10} 18 - \log_{10} 3
\log_{10} 7 - \log_{10} 28
\log_{5} 11 + \log_{5} 2 + \log_{5} 9
\log_{10} 5 + \log_{10} 7 - \log_{10} 3
\log_{7} 12 - \left(\log_{7} 2 + \log_{7} 3\right)
3 \left(\log_{10} 9 + \log_{10} 2\right)
3 \left(\log_{10} 6 - \log_{10} 2\right)
2 \log_{5} 22 - 2 \log_{5} 11
5 \log_{10} 6 + 5 \log_{10} 3
3 + \log_{4} 7
Simplify each of the following expressions in exact form without using a calculator:
\log 4 + \log 9
\log_{10} \left(10\right) + \log_{10} \left(10\right)
\log_{10} 11 + \log_{10} 2 + \log_{10} 9
\log_{10} 12 - \left(\log_{10} 2 + \log_{10} 3\right)
\dfrac{\log_{10} 4}{\log_{10} 2}
\dfrac{\log_{4} 125}{\log_{4} 5}
\dfrac{\log a^{8}}{\log a^{4}}
\dfrac{\log a^{3}}{\log \sqrt[3]{a}}
\dfrac{\log \left(\dfrac{1}{x^{4}}\right)}{\log x}
\log_{10} 10 + \dfrac{\log_{10} \left(15^{20}\right)}{\log_{10} \left(15^{5}\right)}
\dfrac{8 \log_{10} \left(\sqrt{10}\right)}{\log_{10} \left(100\right)}
10^{\log w}
\log 10 x + \log 10 y
x^{ 4 \log_{x} 3 - 6 \log_{x} 2}
y = \log_{a} \left(\sqrt{x} + \sqrt{x - 1}\right) + \log_{a} \left(\sqrt{x} - \sqrt{x - 1}\right)
Given that a > 1, fully simplify the following expressions:
\log_{a} \left(\dfrac{1}{a}\right)
\log_{a} \left(a^{9}\right)
\log_{a} \left(\dfrac{1}{a^{2}}\right)
\log_{a} \left(\sqrt{a}\right)
\log_{a} \left(\dfrac{1}{\sqrt{a}}\right)
Write each of the following as a single logarithm or integer:
5 \log x^{3} - 4 \log x^{2}
5 \log x + 3 \log y
8 \log x - \dfrac{1}{3} \log y
7 \log x - \log \left(\dfrac{1}{x}\right) - \log y
7 \log_{10} 5 - 21 \log_{10} 25
5 \log_{10} 8 - 3 \log_{10} 4
2 \log_{6} 3 + \dfrac{1}{3} \log_{6} 64
\log_{2} 36 - 2 \log_{2} 3
Write \log \left(\dfrac{2 u}{3 v}\right) in terms of \log 2, \log u, \log 3 and \log v.
Express the following as products:
\log_{a} A^{ - 2 }
\log_{6} \sqrt{w}
\log_{p} q^ r
\log_{3} B^ \frac{1}{3}
Rewrite the following as the sum or difference of logarithms without any powers or surds:
\log_{9} u v
\log \left(x^{\frac{2}{5}}\right)
\log \left( 3 x^{ - 1 }\right)
\log \left( 7 x^{ - 4 }\right)
\log \left(\left( 5 x\right)^{ - 7 }\right)
\log \left(\left( 2 x\right)^{ - 1 }\right)
\log \left(\dfrac{1}{x y}\right)
\log \left(\left( 3 x + 7\right)^{ - 1 }\right)
\log \left(\sqrt{\dfrac{c^{8}}{d}}\right)
Rewrite the expression \log x^{2} + \log x^{3} in the form k \log x.
Show that \log_{2} 5 = \dfrac{1}{\log_{5} 2}.
Amy has written the following:
\log_{b} 64 = \log_{b} \left( 64 \times 1\right) = \log_{b} 64 + \log_{b} 1
Is Amy correct? Explain your answer.
Rewrite the following in terms of base 10 logarithms:
\log_{4} 16
\log_{3} 0.9
\log_{3} \sqrt{5}
\log_{a}B
Rewrite \log_{3} 20 in terms of base 4 logarithms.
Use the properties of logarithms to evaluate the following expressions:
\log_{2} 16
\log_{8} \left(\dfrac{1}{64}\right)
\log_{5} 0.2
\log_{4} 1
\log_{36} 6
\log_{2} \left(\dfrac{1}{4}\right)
\log_{10} 0.1
\log_{7} \sqrt[3]{7}
2^{\log_{2} 3}
\log_{2} \sqrt[4]{2}
\log_{3} 3
\log_{16} \sqrt{2}
\log_{2} \left(\sqrt[3]{\dfrac{1}{16}}\right)
\log_{5} 125^{\frac{5}{4}}
Use the properties of logarithms to evaluate the following expressions:
\log_{10} 10^{\frac{5}{4}}
\log_{10} \left(10^{\sqrt{5}}\right)
\log_{16} \sqrt{2}
\log_{10} 2 + \log_{10} 5
\log_{4} 8 + \log_{4} 2
\log_{6} 12 + \log_{6} 18
\log_{2} 72 - \log_{2} 9
\log_{2} 36 - 2 \log_{2} 3
\log_{6} 12 + \log_{6} 15 - \log_{6} 5
\log_{3} 2 - \log_{3} 18
2 \log_{6} 3 + \dfrac{1}{3} \log_{6} 64
Consider the following logarithmic expressions:
Rewrite the expression in terms of base 10 logarithms.
Hence, evaluate each correct to two decimal places.
If \log_{10} 6 = 0.778, calculate \log_{10} \left(\dfrac{1}{216}\right), without using a calculator.
If \log_{a} 3 = 1.16 and \log_{a} 2 = 0.73, find the value of \log_{a} \sqrt{54}, without using a calculator.
If \log_{k} a = 1.64, find the value of \log_{k} k a^{4}.
Using the rounded values \log_{x} 3 = 0.62 and \log_{x} 4 = 0.78, find the value of each of the following expressions:
\log_{x} 9
\log_{x} \sqrt{3}
\log_{x} 4 x
\log_{x} \dfrac{1}{3}
\log_{x} 36
Given that \log_{b} x = 2.6 and \log_{b} y = 4.2, determine the value of the following:
\log_{b} x^{3}
\log_{b} \sqrt[3]{y}
\log_{b} \left( x^{2} \sqrt{y}\right)
\log_{b} \left(\dfrac{b}{x}\right)
If \log_{x} 4 = 3.42 and \log_{x} 12 = 6.13, determine the value of the following:
\log_{x} \left(\dfrac{1}{3 x}\right)
Prove the following properties of logarithms:
\log_{a} x^{n} = n \log_{a} x